arxiv:1608.06362v2 [cond-mat.mes-hall] 10 mar 2017 the monopole defects … · 2018-09-23 · with...

Ultrasonic elastic responses in monopole lattice

Xiao-Xiao Zhang

Department of Applied Physics, The University of Tokyo, 7-3-1 Hongo,Bunkyo-ku, Tokyo 113-8656, Japan

E-mail: [email protected]

Naoto Nagaosa

Department of Applied Physics, The University of Tokyo, 7-3-1 Hongo,Bunkyo-ku, Tokyo 113-8656, JapanRIKEN Center for Emergent Matter Science (CEMS), 2-1 Hirosawa, Wako,Saitama 351-0198, Japan

E-mail: [email protected]

March 2017

Abstract. The latest experimental advances have extended the scenario ofcoupling mechanical degrees of freedom in chiral magnets (MnSi/MnGe) to thetopologically nontrivial skyrmion crystal and even monopole lattices. Equippedwith a spin-wave theory highlighting the topological features, we devise aninteracting model for acoustic phonons and magnons to explain the experimentalfindings in a monopole lattice with a topological phase transition, i.e., annihilationof monopole-antimonopole pairs. We reproduce the anisotropic magnetoelasticmodulations of elastic moduli: drastic ultrasonic softening around the phasetransition and a multi-peak-and-trench fine structure for sound waves paralleland orthogonal to the magnetic field, respectively. Comparison with experimentsindicates that the magnetoelastic coupling induced by Dzyaloshinskii-Moriyainteraction is comparable to that induced by the exchange interaction. Otherpossibilities such as elastic hardening are also predicted. The study implies thatthe monopole defects and their motion in MnGe play a crucial role.

arX

iv:1

608.

0636

2v2

[co

nd-m

at.m

es-h

all]

10

Mar

201

7

Ultrasonic elastic responses in monopole lattice 2

1. Introduction

Topology is now playing a more and more significant role in condensed matter physics.Apart from an upsurge in the focus on topological classification of quantum phasesof matter, another field bearing ideas of topology, magnetism inhabited by stripes,vortices, domain walls, etc., has been under experimental and theoretical investigationsfor a long history[1]. In the recent decade, the realization of topologically nontrivialSkyrmion or Skyrmion crystal (SkX) in chiral magnets[2, 3, 4, 5, 6, 7, 8, 9, 10, 11]revives the old idea originally proposed as a hadron model[12], serving as a newscenario of the interplay between the orbital and spin of electrons and ions, andoffering plenty of brand new phenomena[13, 14, 15, 16, 17, 18, 19, 20, 21, 22] togetherwith the potential for application in magnetic storage[23, 24, 25, 26, 27, 28, 29].Symmetry breaking of spins in noncentrosymmetric chiral magnets, which bears boththe Heisenberg exchange interaction (EXI) and the Dzyaloshinskii-Moriya interaction(DMI)[30, 31, 32] due to spin-orbit coupling, can embody the Skyrmion texture[2, 3].A minimal Hamiltonian for isolated Skyrmions or a Skyrmion lattice in d spatialdimensions includes the EXI, the Bloch-type DMI, and the Zeeman energy[33, 34](setting ~ = 1 throughout this paper)

HSkX =

ˆdd~r

[J

ad−20

(∇~S)2

+D

ad−10

~S ·(∇× ~S

)− 1

ad0µ~S · ~B

](1)

in which a0 is the microscopic lattice constant. In two-dimensional (2D) thin films,usually magnetic anisotropies can also play a key role in stabilizing the Skyrmionphase. Dissimilar to other modulated magnetic structures like the conical and helicalphases that can be realized therein, a Skyrmion winds a sphere certain times and ischaracterized by the topological Skyrmion number[35]. In such magnetic systems, theinvolvement of topology is often enhanced or even induced by strong correlation effect.Indeed, based on an adiabatic approximation for the real-space Berry phase producedby the fixed-length spin texture of Skyrmion, the constraint drawn by the strongcoupling with itinerant electrons can be described by the emergent electromagneticfields (EEMF)[36, 37, 38, 15, 39].

Apart from the most common Skyrmion lattice as vertical magnetic field inducedtriangular lattices of Skyrmions observed in chiral magnets, the coalescence orbisection of columnar Skyrmion tubes has been observed experimentally in a bulkFe1−xCoxSi material[40]. Such merging points are in fact singularities or defects (theBloch points[41, 42]) in the spin texture

~n = ~S/|~S|where the spin moment ~S = ~0. Specifically, they can be regarded as monopolesor antimonopoles[40, 39, 43] in terms of the EEMF and can create or annihilateSkyrmions, which are the tubes in three dimensions (3D). In a Skyrmion latticewith these defects, the Skyrmion tubes may not penetrate the sample and can goto an end at the monopole defects at a depth. For simplicity, we henceforth mayuse ’monopole’ to refer to both monopoles and antimonopoles. A two-Skyrmion-merging model based on the nonlinear sigma model has been used to study its effect ontranport[44]. Another regular solution for such ended Skyrmions has been investigatedin the conical phase[45]. Based on the Landau-Lifshitz-Gilbert equation, people alsostudied the dynamics and energetics of the monopole defects[46, 47].

Partly owing to the intrinsic 3D nature of the monopole defects, there turnsout to be no experimental or theoretical evidence for the formation of a lattice


of the monopoles in thin films[2, 3, 48, 49, 50]. Nevertheless, it should bemore feasible to be realized in 3D bulk chiral magnets. First being theoreticallysuggested as a superposition of multiple non-coplanar spin spirals[51], a 3D latticeof monopole defects found corroboration from the thermodynamic calculationfollowed[52]. Experimentally confirming these expectations, from the analysis ofthe electric and thermal Hall effects and the Nernst effect in contrast to those ofconventional MnSi-like ones, a 3D bulk MnGe under pressure was strongly inferredto possess Skyrmion tubes and a simultaneous lattice of the monopole-antimonopoledefects[53, 54, 55]. Notably, with a lattice variant of the minimal Hamiltonian Eq. (1),latest Monte Carlo simulation confirmed the formation of a monopole-antimonopolelattice in 3D as long as the magnetic period becomes short[56], which is just thesituation of MnGe observed in those experiments. This endorses our inclusion ofthe EXI and DMI in Sec. 2 among other possible magnetic interactions. Further,a recent study using Lorentz transmission electron microscopy clearly revealed themagnetic structure of MnGe in concordance with the three-orthogonal-spiral modelanticipated[57].

It is necessary to pay attention to the distinction between two points of view,i.e., the spin texture ~n(~r) and the original spin moment field ~S(~r), of the magneticordering background, upon which we shall mainly study the effect due to fluctuations.The latter, ~S(~r), which is for constructing the multi-spiral spin density waves, is non-singular and topologically trivial because any configuration mapped onto a 3-ball B3

can be smoothly connected to ~S(~r) = ~0. However, this work aims at highlighting theeffect characteristic for the topologically nontrivial defects and the orientational field~n(~r), which is inherently described by the second homotopy group of a 2-sphere S2

that cannot shrink to a point, naturally appears to be physically more relevant. On theother hand, for MnGe, in the strong correlation regime, localized spins’ moments canhardly vary in magnitude and the saturated magnetization is large and only vanishesat the singular defects. Indeed, this choice of the unit-length field already proved to beappropriate for the description of the influence on conduction electron in the stronglycorrelated MnGe material, where the 3D monopole lattice even exists in the absenceof external magnetic field[43, 39].

Expecting new effects due to the monopole defects, we initiated our theoreticalinvestigation to study the phenomena emerging from the coupling between othercomponents and the topologically nontrivial monopole lattice in the hosting materialMnGe by calculating the influence from spin-wave excitations therein. Our first studyfocuses on the longitudinal electric transport in MnGe massively influenced by thequantum fluctuations of the EEMF due to spin waves and identified a topologicalphase transition of strong correlation genesis[39]. One more intriguing possibilityis the coupling to phonons in the hosting solids. Indeed, unconventional ultrasonicresponses of elastic stiffness have been observed recently. Experimentalists employedlongitudinal sound waves to the Skyrmion phases in both MnSi[58, 59] and MnGe[43]with the sound propagating direction parallel (k‖-mode) or orthogonal (k⊥-mode)to the applied external magnetic field along the z-axis as shown in Fig. 1. Theyrecognized the SkX phase by a distinct elastic stiffness with anisotropy for the twomodes in MnSi. Dissimilarly, varying with external magnetic field, MnGe showed notonly a drastic softening of k‖-mode stiffness but also a multi-peak-and-trench finestructure of k⊥-mode stiffness, which are much stronger than those in the SkX phaseof MnSi. The range of the magnetic field for the softening considerably coincideswith the topological phase transition aforementioned, which is the destruction of the


(a) k‖-mode (b) k⊥-mode

Figure 1. Illustration of experimental settings for the k‖-mode and k⊥-mode. kand H stand for the sound wave and the external magnetic field, respectively.

monopole lattice. Fluctuations associated with such qualitative structural transitionis therefore not surprising to produce dramatic modification in mechanical properties.The EEMF not only helps describe itinerant electrons coupled with localized spinsbut also captures the essential feature of the nontrivial spin texture, hence we findit rather informative even for the magnetoelastic response to be discussed. And itis reasonable to owe the qualitatively different responses of MnGe compared withMnSi to the influence from the monopoles/antimonopoles on the magnon fluctuationstherein.

Towards this end, we reuse our spin-wave theory and propose in this paper aninteraction theory for magnons and phonons induced by both EXI and DMI, whichreproduces and explains the magnetoelastic experimental features. We refer thereader to the separate papers[39, 43] for the detailed description of the monopolelattice consisting of three orthogonal spin spirals, the spin-wave theory of a Skyrmionlattice or monopole lattice, and the experimental confirmation. In Sec. 2, we firstdevise the magnetoelastic interactions and derive the effective phonon theory byintegrating out the magnon degrees of freedom. The influence of the magnons onthe renormalized phonon excitations is discussed in Sec. 3.1 with an emphasis onthe renormalized phonon linear mode. In Sec. 3.2 and Sec. 3.3, we compare theexperimental observations with our theoretical predictions and explain the physicalorigin of the magnetic-field-dependent evolution of the hybridized excitation spectra.We emphasize the distinction between the monopole lattice in MnGe and theconventional SkX in MnSi in the rest of Sec. 3 and conclude in Sec. 4.

2. Theoretical Models

2.1. Magnetoelastic coupling

As aforesaid, it is the unit-norm constrained spin texture ~n = ~S/|~S| that produces theEEMF and yields the nontrivial topology of the Skyrmion and the monopole ratherthan the bare spin moment ~S itself, which manifests the strong electron correlationtherein[39, 43]. If ~S were used, the form factors to be calculated in the following wouldassume too simple forms to give rise to the complex enough magnetic-field dependenceand the correct theory to reproduce experiments. We thus make use of ~n instead of ~Sto derive the appropriate magnetoelastic interactions from the Hamiltonian Eq. (1).This is the minimal model capable of producing a stable monopole lattice as discussedpreviously. And this study shows that EXI and DMI are adequate and essential tocapture the correct physics herein.

When longitudinal sound waves are artificially generated in the material, the


Figure 2. Cartoon for the mechanism of the magnetoelastic coupling.

lattice structure will be perturbed, which is described by the longitudinal phonon.However, the strengths of EXI and DMI depend on lattice bond lengths hence themagnetoelastic coupling (Fig. 2). Such a perturbation to the magnetic interactionsby mechanical degrees of freedom can be accounted for by expanding the strengths ofEXI and DMI up to the linear order of phonon degrees of freedom

J(∇~n)2 → (J0 + αEXI(∂juj))∂jni∂jni (2a)

D~n · (∇× ~n)→ εijk (D0 + αDMI(∂juj))ni∂jnk (2b)

wherein ~u is the lattice displacement from equilibrium in the continuum limit,αEXI/DMI is the coefficient of this expansion, and summation over repeated indicesis understood henceforth. We emphasize four features of this expansion. 1) Inconformity with translational symmetry, either J or D depends on ∂juj rather thanlattice displacement ~u itself. 2) Alongside, ∂juj obviously deals with the longitudinalphonon. 3) All spatial derivatives match in the direction since any ∂j~n, reflectingmagnetic interactions between two adjacent spins sitting at the endpoints of a latticebond along the j-direction in a lattice model, should be affected by the longitudinalphonon propagating along the same direction through ∂juj . 4) As stated in 3), thesemagnetoelastic couplings take anisotropy into account so as to produce the delicateexperimental observations.

The next step is to expand ~n as ni = n(0)i + ϕµ∂ϕµn

(0)i in Eqs. (2a)(2b) to

incorporate the spin-wave degrees of freedom ϕµ = (~φ , ~δm) , µ = 1, · · · , 6 and thesuperscript (0) means the ground state value or setting ϕµ = 0 after taking thederivative. For a spin spiral indexed by i, this φi is the phase of the constituentspin density wave, which also indicates the spatial shift of the Skyrmion lattice or theposition of the monopole defects. And mi is the magnetization along the propagationaxis of spin spiral i, generating rotation around this axis. Note that we considerthe situation where there are three orthogonal spin spirals and it is those quantities’deviations away from their static mean field values that constitute the spin-wave fieldsϕµ in the expansion[39]. Without loss of generality, we thus set the static value of anyφi to 0 and denote the fluctuation in mi by δmi.

Starting from the EXI and DMI induced magnetoelastic interaction parts inEqs. (2a)(2b), up to terms bilinear in spin-wave and phonon fields, one arrives at

EXI part:[∂ϕµ(∂jn

(0)i ∂jni(0))ϕµ + 2∂ϕµn

(0)i ∂jni(0)∂jϕµ

]∂juj (3a)

DMI part: εijk[∂ϕµ(n

(0)i ∂jn

(0)k )ϕµ + n

(0)i ∂ϕµn

(0)k ∂jϕµ

]∂juj , (3b)

wherein we temporarily omit all the prefactors for simplicity. One crucial criterion isthe translational invariance in the continuum model to be derived, i.e., for the spin-wave field φi, only its spatial derivative can enter simply because φi is the (phase)displacement field of spin spiral i. Therefore, when µ = 1, 2, 3, i.e., for the φi field,


∂ϕµ applied upon the parenthesis in the first term in either Eq. (3a) or Eq. (3b) isequivalent to a spatial derivative. And we notice that the typical wave vector of theperiodic function in the parenthesis is much larger than that of the slowly varyingspin-wave field φi. Since Eqs. (3a)(3b) will be part of a Lagrangian density, a spatialintegration can be carried assuming φi is uniform, resulting in zero in the end forthe total derivative. This procedure exactly eliminates the term linear in φi field,otherwise the translational invariance would be violated. However, this is unnecessaryfor µ = 4, 5, 6, i.e., for the fluctuating field δmi. Therefore, in either Eq. (3a) orEq. (3b), for µ = 1, 2, 3, only the second term remains.

Thus for either EXI or DMI case, we attain two terms

Cµj(∂jϕµ∂juj) +Dµj(ϕµ∂juj) (4)

in total, wherein µ = 1, . . . , 6 but Dµj ≡ 0 whenever µ < 4. We call thefunctions preceding the bilinear fields form factors, which render the couplingbetween magnon and phonon nonuniform in space. They are defined as CµjEXI =

2∂ϕµn(0)i ∂jni(0) , Dµj

EXI = ∂ϕµ(∂jn(0)i ∂jni(0)) , CµjDMI = εijkn

(0)i ∂ϕµn

(0)k , Dµj

DMI =

εijk∂ϕµ(n(0)i ∂jn

(0)k ) and actually record the information of the complicated magnetic

structure affecting the EXI or DMI induced magnetoelastic interactions. After Fouriertransform to the momentum space, we take the spatial average over a magneticunit cell of the form factors, making the couplings dependent only on the variableuniform magnetization mz along the z-axis. Therefore we have an exactly solvabletheory without couplings between unequal momenta. This uniform magnetization mz

directly relates to the external magnetic field applied on the system along the z axis.The magnetic field dependence of the ultrasonic responses is due to the fact that themagnetic texture of the monopole lattice varies with respect to mz[39].

2.2. Effective theory of phonon

A standard free theory of the longitudinal phonon reads

Lph =1

2

[c(∂τ~u)2 + κ(∇ · ~u)2

], (5)

wherein κ is an elastic constant, c is the mass density and we work in imaginary timehenceforth. For MnGe, spin-wave LSW takes the form[39]

LSW =∑

α=x,y,z

[iεαβγAbαφβφγ − iBδmαφα + χδmα

2 + ρ(∇φα)2]

(6)

wherein A = −2qeS1

kjkk1ad0, B = 1

ad0, χ = D2

Jad0, ρ = J

ad−20

. Note that the mz-dependent

~b is the emergent magnetic field that characterizes the nontrivial topology of theSkyrmions or monopoles in the system. And we get the magnetoelastic interactions

LME =αEXI

ad−20

(CµjEXI∂jϕµ∂juj+DνjEXIϕν∂juj)+

αDMI

ad−10

(CµjDMI∂jϕµ∂juj+DνjDMIϕν∂juj)(7)

from Sec. 2.1. Because the three parts, Eqs. (5)(6)(7), comprising the full theory are allbilinaear, we can diagonalize the actions in the energy-momentum space. Below, k isused as a shorthand for the argument (~k, z), wherein z is a generic complex frequencythat can equal iωn for instance. The longitudinal phonon action is transformed to

Sph =

ˆ β

0

dτ

ˆdd~rLph =

1

2

∑k

uT(k)Mph(k)u(−k),


in which 3× 3 matrix (Mph)ij = κkikj − cz2δij and u = (ux, uy, uz)T. The spin-wave

action is transformed to

SSW =

ˆ β

0

dτ

ˆdd~rLSW =

1

2

∑k

ϕT(k)MSW(k)ϕ(−k),

in which 6 × 6 matrix MSW can be solved and ϕ = (ϕ1, . . . , ϕ6)T. The action ofmagnetoelastic interactions becomes

SME =

ˆ β

0

dτ

ˆdd~rLME =

∑k

FT(k)ϕ(−k),

wherein 6-vector F (k) = U(k)u(k) = (UEXI(k) + UDMI(k))u(k), 6 × 3 matrices(UEXI)

µj = αEXI

ad−20

[(kj)2CµjEXI + ikjDµjEXI] , (UDMI)

µj = αDMI

ad−10

[(kj)2CµjDMI + ikjDµjDMI].

Now the full theory is constituted of three parts S = Sph + SSW + SME, whichare the free phonon theory, the free magnon theory, and their coupling due to themagnetoelastic interactions in turn. Next, we can integrate out the harmonic magnonbath in the path-integral formalism, leading to the partition function expressed as

Z[ϕ, u] =

ˆDϕDu e−S = Z[ϕ, u ≡ 0]

ˆDu e−Seff .

Because of the bilinearity, after a Gaussian functional integration, we get the effectiveaction for the phonons

Seff =∑k

uT(k)

[−1

2UT(k)M−1

SW(k)U(−k) +1

2Mph(k)

]u(−k). (8)

The expression in the bracket is nothing but −1 multiplying the inverse of the 3 × 3Matsubara Green’s function matrix Geff(k) of the effective phonon theory in the energy-momentum space.

3. Methods and results

3.1. Renormalized phonon spectrum

Instead of the hardly accessible analytic dispersion relations, we resort to inspectingthe spectral function A(~k, ω) of this effective theory. In conformity with experimentalinvestigations, we focus on the effects due to sound waves propagating along x and zdirections by looking at the diagonal Aii(ki, ω) = −2=GR

ii as a function of momentumki along i-direction and frequency ω, wherein retarded Green’s functionGR

ii comes fromthe analytic continuation (Geff)ii (ki, z → ω+iη). First of all, we confirm the propertyof the bosonic spectral function[60] that it is always positive for ω > 0 and negativefor ω < 0. Secondly, the renormalized phonon spectrum is reflected in Aii(ki, ω) plotby δ-function-type ridge structures. Thirdly, setting a realistic but small enough η,we can extract the information of the phonon excitations from the Aii(ki, ω) plot byidentifying the ridge structures.

Equation (5) itself can only give banal longitudinal phonon excitations of lineardispersion ω = v0k with the acoustic velocity v0 =

√κ/c. On the other hand,

the magnon theory possesses an excitation spectrum composed of three distinctmodes[39]. For MnGe, there are two kinds of gapless excitation, one acoustic modeω1 = 2Da0k ∝ Dk and one quadratic mode ω2 = ρ

Abk2 ∝ Jk2 when k is small, and

another excitation with an energy gap ∆mag = 4χAbB2 ∝ D2

J , as shown in Fig. 3.


Figure 3. Low-energy magnon spectrum.

Because of the spin Berry phase of the topologically nontrivial spin textures ofSkyrmions/monopoles, any two unparallel spin spirals are possible to mingle witheach other. This gives rise to φ-quadratic couplings like φxφy in Eq. (6), mixing upconjugate φ fields and hence the quadratic mode[39, 15, 61]. An exception is that thesethree modes degenerate into the linear gapless mode ω1 when mz = 0. These gaplessexcitations are the Nambu-Goldstone bosons due to the broken translation symmetry.In the case of electric transport[39], the softest one affects the electrons’ motion themost. However, all the gapless modes will play a crucial role in the low-energy physicsdescribing the interplay with the linear acoustic phonons.

The salient point is that because of the magnetoelastic coupling, the threemagnon modes will hybridize with the phonon mode, giving rise to rich possibilitiesof renormalized excitation spectra. One interesting instance occurs when the phononmode intersects with the magnon modes in the dispersion plot, resulting in mutualrepulsion and reconstruction of the involved dispersion curves. In the gross, despitepossible reconstructions, one is in general still able to relate the new modes to theirrespective precedents before hybridization, which will henceforth be used as convenienttags of the new modes in the effective phonon theory. Even without any intersectionin the dispersion curves, the intensity of a new magnon mode (sharpness of the ridgestructure) in a spectral function plot directly reflects the degree of hybridizationthat influences the (approximately) linear renormalized phonon mode. We have therelation Vrenorm(mz) =

√κrenorm(mz)/c (seen from Eq. (5)) between the velocity of

the renormalized phonon linear mode and the new stiffness κrenorm. This phononlinear mode after hybridization is of the major importance because it is this stiffnessκrenorm that is measured experimentally at different external magnetic fields as theultrasonic responses. Its velocity is the slope of the corresponding dispersion ridgeextracted from scanning the spectral function, e.g., in the shaded region in Fig. 4.

Because of the experimental difficulty in determining the accurate value of J , Dand the magnetoelastic couplings αEXI/DMI, one has to search for the correct controlparameters corresponding to the real materials. We consider two, αDMI

EXIand Rvelo,

for the effective theory. One is the ratio between strengths of different magnetoelasticcouplings αDMI

αEXI, denoted as αDMI

EXI. The other is the ratio of the velocity of the magnon

linear mode, vmag = 2Da0, to the original unperturbed acoustic velocity v0, denotedas Rvelo =

vmag

v0= 2Da0√

κ/c. We also give an estimate of the characteristic energy scales

in the MnGe experiment. The sound wave propagating in MnGe crystal has frequency


Figure 4. Scanning the shaded region to extract the renormalized phononvelocity from a typical logarithmic plot of the spectral function Axx(kx, ω).

f0 = 18MHz and wavelength λ = 260µm[43]. The original acoustic velocity is hencev0 = λf0 = 4.7 × 103m/s and the phonon energy is ε0 = ~ω0 = 1.2 × 10−26J. As forthe material MnGe[54, 57], a0 is about 4A and the melting temperature (∼ 200K)of the magnetic order can be used to estimate the strength of EXI J . We setJ = 10D for MnGe. Thus, it is straightforward to obtain vmag = 12.0 × 103m/sand ∆mag = 2.4× 10−23J, using the typical value when mz = 0.8. On the other hand,in our theoretical study, we set η = 1 × 10−6, aSkX = 2π, ~ = κ = c = 1 and hencev0 = 1. Consequently, J or D is fixed by Rvelo. In addition to the foregoing variableαDMI

EXI, we set αEXI = 0.8 since a too large magnetoelastic coupling unrealistically alters

the phonon spectrum while a tiny one renders the effect feeble to detect.The first message from the above is that vmag is of the same order as v0. This

implies that we had better tune Rvelo not far from unity if we were to explain theexperiment. The second message is ε0 � ∆mag, which means the low-energy phononexcitations and hence gapless magnons around the long wavelength limit play a majorrole. Because the U matrix in Eq. (8) contains two individual parts due to EXI andDMI, the combination becomes not simply a summation of the separate effects, butone with inevitable interference between the two types of magnetoelastic interactions.Indeed, we have seen distinct Vrenorm(mz) profiles when changing αDMI

EXIwithin a typical

range [−5, 5] by step 0.1. Note that the sign difference between two magnetoelasticinteractions is possible as implied by the sign change in DMI[62, 63, 64].

3.2. Rich possiblities of magnetoelastic responses

The value of Rvelo strongly affects the excitation spectrum of the effective phonontheory in a clearer manner as compared with αDMI

EXI. Imagine drawing the original

unperturbed phonon dispersion line in Fig. 3, whose slope might be smaller orlarger than that of the blue magnon linear mode, providing a natural bipartiteclassification. 1) v0 < vmag. The phonon mode lies below the blue magnon linearmode and intersects with the green magnon quadratic mode. On one hand, if both themagnon quadratic mode itself and the magnetoelastic interactions are strong enoughin intensity, depending on the details of the coupling, the intersection becomes ananticrossing of two reconstructed modes repelling each other in the effective phononspectrum. On the other hand, near the structural phase transition where fluctuationsbecome immense, the blue magnon mode strongly repels the phonon mode downward


0.5 1.0 1.5 2.0 2.5 3.0mz

0.90

0.92

0.94

0.96

0.98

1.00

Vrenormx

(a) k⊥-mode0.5 1.0 1.5 2.0 2.5 3.0

mz

0.90

0.92

0.94

0.96

0.98

1.00

Vrenormz

(b) k‖-mode

Figure 5. The elastic softening. The velocities of the renormalized linearphonon excitations v.s. the uniform magnetization mz under control parametersαDMI

EXI= 1 , Rvelo = 2.4. Important experimental features, strong anisotropy,

substantial softening, and multi-peak-and-trench fine structure, are theoreticallyreproduced.

60K

80K

100K

2 4 6 8H(T)

0.005

0.010

0.015

Δκx/κx(0T)

(a) k⊥-mode

60K

80K

100K

2 4 6 8H(T)

-0.08

-0.06

-0.04

-0.02

0.02

Δκz/κz(0T)

(b) k‖-mode

Figure 6. The experimentally observed softening signal at several temperatures.The relative change of the elastic constant κx/z v.s. the external magnetic fieldH along z direction. Characteristic trench and peak positions are marked by graydots. Figure plotted from experimental data which are partly reported in theseparate paper[43].

as a result of the magnon-phonon interaction, serving as the major cause of thesoftening effect in Fig. 5 in agreement with the experimental data shown Fig. 6. 2)v0 > vmag. The phonon mode lies above the blue magnon linear mode and intersectswith the orange magnon gapped mode. In spite of this, the high energy scale of thegap renders itself irrelevant for the long wavelength phonons. Similar to 1), near thephase transition, the blue mode repels the phonon mode upward, which is an elastichardening prediction from our theory. As seen in Fig. 7, the renormalized phononvelocity is in general larger than its original unity value. It is expected to be realizedby changing either v0 or vmag. Recent studies of controllable DMI in Skyrmion latticesby exerting strains[62, 65] or varying compositions[64] can be candidates for realizingthis case. Last but not least, when v0 ≈ vmag, the foregoing distinction becomesvague. As a minor reassurance, we indeed saw evident, albeit complex transitionsfrom softening to hardening in Vrenorm(mz) when Rvelo traverses the range [0.7, 1.3].

We discuss more details about the comparison between theory and experiment.Figure 5 shows the theoretical result that reproduces the experimentally observed


0.5 1.0 1.5 2.0 2.5 3.0mz

1.00

1.05

1.10

1.15

1.20

Vrenorm

x

0.5 1.0 1.5 2.0 2.5 3.0mz

1.00

1.05

1.10

1.15

1.20

Vrenorm

z

(a) αDMIEXI

= −1 , Rvelo = 0.2

0.5 1.0 1.5 2.0 2.5 3.0mz

1.05

1.10

1.15

1.20

Vrenorm

x

0.5 1.0 1.5 2.0 2.5 3.0mz

1.05

1.10

1.15

1.20

Vrenorm

z

(b) αDMIEXI

= 2 , Rvelo = 0.6

Figure 7. The elastic hardening. Vx/zrenorm(mz) plots of Rvelo < 1 and different

signs of αDMIEXI

.

signals in Fig. 6, including the drastic decrease in the velocity of the k‖-mode andthe multi-peak-and-trench fine structure for the k⊥-mode. In this model calculation,we cannot produce a temperature dependence simply because the spectral functionin this case doesn’t depend on temperature. However, as suggested by basicallythe same feature at different temperatures marked by the gray dots in Fig. 6, thecurrent theory is able to illustrate the essential physics therein. The external magneticfield in Fig. 6 is in general not simply proportional to the parameter mz in Fig. 5since mz should be regarded as an approximation in modeling the effect on thedeformation of the magnetic structure due to the possibly complex magnetizationprocess. Nevertheless, this comparison suffices to highlight the key magnetoelasticresponses. The reliability of this result is supported by the fact that within therange αDMI

EXI∈ [0.8, 1.2] , Rvelo ∈ [1.6, 4] of the control parameters, the basic characters

hold all along. After all, we can notice that in Fig. 5 the most evident changes inthe velocity (stiffness) always reside around the destruction of the monopole lattice,i.e., the monopole-antimonopole pair annihilation at mz =

√2[39], and especially this

topological phase transition clearly manifests itself by the drastic softening in Fig. 5(b).As for the magnitude of the softening with respect to the original v0 = 1 situation,

our theory produces∆V zrenorm

∆V xrenorm. 5, which is a bit smaller than the experimental value

between 6 and 10. Despite this discrepancy, we highlight the excellent consistencybetween the theoretical and experimental fine structures. Besides the clear match forsingle-trench k‖-mode case of Fig. 5(b), all three trenches and two peaks in Fig. 5(a)find their counterparts in the experiment. On the other hand, if we set αEXI or αDMI

to zero, it becomes impossible to reproduce the experimental signals. Taken as awhole, these strongly endorse our theory and can be regarded as a new way to findout some quantities temporarily outside the reach of experimental detection.

Another aspect worth discussing is why the experimental features are reproducedwhen αDMI

EXIis not far away from unity although we have J = 10D. The resolution

consists in DMI’s unique sensitivity to minute strains[62, 63, 65]. Some anticrossing


Figure 8. Logarithmic plots of spectral function Azz(kz , ω) at various uniformmagnetization mz ’s for the k‖-mode sound wave.

points mixing up spin-up and spin-down bands in the band structure contributemassively to the spin-orbit interaction hence the DMI. The phonon induced strainmodifies the band structure slightly, which, however, may considerably change theDMI because the position of the Fermi level in regard to the nearby anticrossing pointscan drastically change the contribution from these points. Experimentally, typicallyten times larger relative modulation of DMI than that of EXI is observed[65], whichis consistent with our theoretical finding.

3.3. Magnetic-field-dependent evolution of the hybridized excitation spectra

Obviously, noticing the range of Rvelo, Fig. 5 belongs to the foregoing v0 < vmag case.A careful inspection of the dispersion profiles, i.e., spectral function plots Fig. 8 (k‖-mode) and Fig. 9 (k⊥-mode), leads us to the following explanation. For the k‖-mode,the dominant modes appear to be the linear excitations of magnons (upper branch) andphonons (lower branch). Increasing mz, the slope of the magnon mode almost remainsthe same while its intensity gradually increases (mz = 0.6–1.46) to some maximumvalue around the phase transition and drops down afterward (mz = 1.6–2.0). Thelarger the intensity becomes, the stronger repulsion is exerted to the phonon modeunderneath, explaining the drastic softening. The reason why this case lacks theparticipation of the magnon quadratic mode lies in the magnetoelastic interaction inEq. (4). The magnon quadratic mode originates from the φ-quadratic term in ourspin-wave theory in Eq. (6). Thus, if it were to largely affect the phonon linear modein the effective phonon theory, φ-field must have an adequate coupling with phonons.


The relevant form factors hereof, CµjEXI/DMI when µ = 1, 2, 3, is diagonal as seen from

the calculation, making the k‖-mode (k⊥-mode) phonon primarily couples with φz(φx). But only φx and φy in the φ-quadratic term are important since bx and by arevanishing.

For the k⊥-mode, the interplay takes place mainly between the original phononlinear mode and the magnon linear and quadratic modes as seen from their highintensities in Fig. 9. We can first faintly recognize the three magnon modes (gapped,linear and quadratic, from left to right) together with the always dominant phononlinear mode when mz = 0.02, which is in contrast to the mz = 0 case with onlyone magnon linear dispersion present. This is because of the degeneracy of magnonexcitations when mz = 0 mentioned in Sec. 3.1. Note that the gapped magnonmode disappears in most plots since increasing mz enlarges the gap beyond the plotrange. For mz = 0.02 and mz = 0.1, a typical (anti)crossing or reconstruction ofthe magnon and phonon linear modes occurs at some low-energy scale inside the plotrange. The mutual repulsion between them makes the reconstructed phonon linearmode, especially the part right to the crossing point, move downward, explaining thefirst small trench in Fig. 5(a). Once the crossing point gets higher (mz = 0.3), thelow-energy part of the phonon mode simply bounces back and gives rise to the firstsmall peak near mz = 0.25 in Fig. 5(a). Thereafter, as mz increases till the vicinityof 1.2, the magnon linear mode above the phonon linear mode becomes stronger andstronger while the magnon quadratic mode keeps moving down till mz = 1.1, whichconstructively pushes the phonon mode downward, yielding the deepest trench. Andthen, abruptly, a transient reverse procedure is observed approximately from mz = 1.1to mz = 1.42 plots, hence the dramatic upsurge in Fig. 5(a). Next, the magnonquadratic mode recovers all the way back and crosses over the phonon mode at lowerand lower energy scales while the magnon linear mode becomes stronger till mz = 1.55,which is again a constructive effect of dragging downward the phonon mode. Finally,a fading reunion of the three magnon modes are observed from mz = 1.7 to mz = 2.2,which is natural for an induced ferromagnetism of too large mz.

In a nutshell, the experimentally observed magnetoelastic phenomena are theconsequences of two aspects that vary with mz or the external magnetic field. Oneis the magnon quadratic mode generated at nonzero mz (together with the hereininsignificant gapped mode), whose reciprocating shift in the spectral function plotin Fig. 9 is clearly controlled by the emergent magnetic field bz(mz) as explained inSec. 3.1 in terms of Eq. (6). The other is the family of form factors directly affecting theintensities of and hybridization between various modes. They show nonmonotonousbehavior upon increasing mz and vary strongly near the phase transition. An intricateintegration of these two aspects leads to the rich experimental features. A detailedinspection of the underlying ground state spin configuration provides more insights[39].Here we only recapitulate two key aspects. One is the nonmonotonic profile of bz(mz),whose maximum near mz = 1.0 and fast dip around mz =

√2 make the form factors’

variation more perceivable. Note that the realistic lattice cutoff introduced to themonopole defects in the calculation will also postpone the complete destruction of themonopole lattice to some value larger than the ideal value mz =

√2. The other, from a

more intriguing viewpoint, is the nontrivial contribution from the (anti)monopoles andtheir characteristic collision-and-annihilation motion. In contrast to the SkX in MnSidiscussed below, we have argued for the crucial role of the topological defects, i.e,the (anti)monopoles, in the magnetoresistivity and especially the topological phasetransition of the destruction of the monopole lattice. Here in the magnetoelastic


Figure 9. Logarithmic plots of spectral function Axx(kx, ω) at various uniformmagnetization mz ’s for the k⊥-mode sound wave.

phenomena, not only bz but also the form factors are reflecting the rich facets of thespin textures. It is the coupling between phonons and the (anti)monopoles that causesall the complexities.

3.4. Relation to triangular lattice of Skyrmion tubes in MnSi

Now we briefly discuss the magnetoelastic couplings in the SkX of MnSi. Except fromthe modification in the spin-wave theory, we can apply the similar form of couplings toMnSi. Nevertheless, we observe vanishing couplings when the sound wave propagatesalong the cylindrical symmetry z axis of Skyrmion tubes in MnSi. For the EXI inducedcoupling, the form factors are simply zero due to the translational symmetry alongthe z direction. Although this is not the case for the coupling induced by DMI, thespatial averages of corresponding form factors turn out to be zero in the end. As for theperpendicular propagating case, we observe nonvanishing magnetoelastic effects, albeitnegligibly smaller as compared to MnGe. On the other hand, the experimental signals


(∆κκ ≈ 0.1%) of the SkX in MnSi are quite smaller than those (∆κ

κ ≈ 2% ∼ 10%)detected in MnGe[43]. Therefore, our form of magnetoelastic couplings turns outto be the leading order effect in the MnGe monopole lattice. Based on the solidstate mechanics and macroscopic thermodynamics, many more complex higher orderterms and fitting parameters are incorporated into a Ginzburg-Landau free energycalculation[66] for the ultrasonic signals in MnSi[58, 59]. This comparison actuallylends credence to the aforementioned essential role played by the monopole defectsthat are absent in MnSi. In other words, the contribution from monopole defects tothe magnetoelastic effects, if present at all, dominates and is captured by the formalismdeveloped in this work.

4. Concluding remark

The study of ultrasonic elastic responses possesses strong motivation from theexperimental findings. We not only explain the observed softening effect but alsopredict new issues, e.g., hardening and other patterns of the dependence of the stiffnesson the magnetic field, which in return provides a way to determine some experimentallyinaccessible physical quantities. Based on a well-established spin-wave theory from theprevious magnetoresistivity study, we are able to identify once again the nontrivialfeatures of the monopole lattice, especially the topological phase transition signifyingstrong correlations. Thanks to the agreement with the experimental observations, thismagnetoelasticity study, together with the magnetoresistivity one, can establish theimportance of the topological nature of the spin configuration in strongly correlatedelectronic systems. As a whole, they speak for a crucial role played by the monopoledefects in chiral magnet MnGe. In particular, these studies pave the way for even moreintriguing scenarios of coupling topologically nontrivial objects with other systems.They show the rich physics therein and help us gain insights for further investigationstowards plenty of manipulation methods for Skyrmionics applications.

Acknowledgments

We thank Yoichi Nii for useful discussions and the indispensable experimental results.X.-X.Z was partially supported by the Panasonic Scholarship and by Japan Societyfor the Promotion of Science through Program for Leading Graduate Schools (ALPS)and Grant-in-Aid for JSPS Fellows (No. 16J07545). This work was supported byJSPS Grant-in-Aid for Scientific Research (No. 24224009) and JSPS Grant-in-Aidfor Scientific Research on Innovative Areas (No. 26103006) from MEXT, Japan, andImPACT Program of Council for Science, Technology and Innovation (Cabinet office,Government of Japan). This work was also supported by CREST, Japan Science andTechnology Agency.[1] A. P. Malozemoff. Magnetic Domain Walls in Bubble Materials: Advances in Materials and

Device Research. Academic Press, 1 1979.[2] A. N. Bogdanov and D. A. Yablonskii. Thermodynamically stable ”vortices” in magnetically

ordered crystals. the mixed state of magnets. JETP, 95:182, 1989.[3] U. K. Roßler, A. N. Bogdanov, and C. Pfleiderer. Spontaneous skyrmion ground states in

magnetic metals. Nature, 442(7104):797–801, Aug. 2006.[4] S. Muhlbauer, B. Binz, F. Jonietz, C. Pfleiderer, A. Rosch, A. Neubauer, R. Georgii, and P. Boni.

Skyrmion lattice in a chiral magnet. Science, 323(5916):915–919, 2009.[5] Akira Tonomura, Xiuzhen Yu, Keiichi Yanagisawa, Tsuyoshi Matsuda, Yoshinori Onose, Naoya

Kanazawa, Hyun Soon Park, and Yoshinori Tokura. Real-space observation of skyrmion


lattice in helimagnet MnSi thin samples. Nano Letters, 12(3):1673–1677, 2012. PMID:22360155.

[6] X. Z. Yu, N. Kanazawa, Y. Onose, K. Kimoto, W. Z. Zhang, S. Ishiwata, Y. Matsui, andY. Tokura. Near room-temperature formation of a skyrmion crystal in thin-films of thehelimagnet FeGe. Nature Materials, 10(2):106–109, Dec. 2010.

[7] X. Z. Yu, Y. Onose, N. Kanazawa, J. H. Park, J. H. Han, Y. Matsui, N. Nagaosa, and Y. Tokura.Real-space observation of a two-dimensional skyrmion crystal. Nature, 465(7300):901–904,Jun. 2010.

[8] W. Munzer, A. Neubauer, T. Adams, S. Muhlbauer, C. Franz, F. Jonietz, R. Georgii, P. Boni,B. Pedersen, M. Schmidt, A. Rosch, and C. Pfleiderer. Skyrmion lattice in the dopedsemiconductor Fe1−xCoxSi. Phys. Rev. B, 81:041203, Jan. 2010.

[9] S. Seki, X. Z. Yu, S. Ishiwata, and Y. Tokura. Observation of skyrmions in a multiferroicmaterial. Science, 336(6078):198–201, 2012.

[10] Stefan Heinze, Kirsten von Bergmann, Matthias Menzel, Jens Brede, Andre Kubetzka, RolandWiesendanger, Gustav Bihlmayer, and Stefan Blugel. Spontaneous atomic-scale magneticskyrmion lattice in two dimensions. Nature Physics, 7(9):713–718, 2011.

[11] Stefan Heinze, Kirsten von Bergmann, Matthias Menzel, Jens Brede, Andre Kubetzka, RolandWiesendanger, Gustav Bihlmayer, and Stefan Blugel. Spontaneous atomic-scale magneticskyrmion lattice in two dimensions. Nature Physics, 7(9):713–718, Jul. 2011.

[12] T.H.R. Skyrme. A unified field theory of mesons and baryons. Nuclear Physics, 31:556 – 569,1962.

[13] Minhyea Lee, W. Kang, Y. Onose, Y. Tokura, and N. P. Ong. Unusual hall effect anomaly inMnSi under pressure. Phys. Rev. Lett., 102:186601, May 2009.

[14] A. Neubauer, C. Pfleiderer, B. Binz, A. Rosch, R. Ritz, P. G. Niklowitz, and P. Boni. Topologicalhall effect in the A phase of MnSi. Phys. Rev. Lett., 102:186602, May 2009.

[15] Jiadong Zang, Maxim Mostovoy, Jung Hoon Han, and Naoto Nagaosa. Dynamics of skyrmioncrystals in metallic thin films. Phys. Rev. Lett., 107:136804, Sep. 2011.

[16] Karin Everschor, Markus Garst, Benedikt Binz, Florian Jonietz, Sebastian Muhlbauer, ChristianPfleiderer, and Achim Rosch. Rotating skyrmion lattices by spin torques and field ortemperature gradients. Phys. Rev. B, 86:054432, Aug. 2012.

[17] R. Ritz, M. Halder, M. Wagner, C. Franz, A. Bauer, and C. Pfleiderer. Formation of a topologicalnon-fermi liquid in MnSi. Nature, 497(7448):231–234, May 2013.

[18] Wei Li, Chiming Jin, Renchao Che, Wensen Wei, Langsheng Lin, Lei Zhang, Haifeng Du,Mingliang Tian, and Jiadong Zang. Emergence of skyrmions from rich parent phases inthe molybdenum nitrides. Phys. Rev. B, 93:060409, Feb 2016.

[19] H Velkov, O Gomonay, M Beens, G Schwiete, A Brataas, J Sinova, and R A Duine.Phenomenology of current-induced skyrmion motion in antiferromagnets. New Journal ofPhysics, 18(7):075016, 2016.

[20] F. Jonietz, S. Muhlbauer, C. Pfleiderer, A. Neubauer, W. Munzer, A. Bauer, T. Adams,R. Georgii, P. Boni, R. A. Duine, K. Everschor, M. Garst, and A. Rosch. Spin transfertorques in MnSi at ultralow current densities. Science, 330(6011):1648–1651, 2010.

[21] X.Z. Yu, N. Kanazawa, W.Z. Zhang, T. Nagai, T. Hara, K. Kimoto, Y. Matsui, Y. Onose, andY. Tokura. Skyrmion flow near room temperature in an ultralow current density. NatureCommunications, 3:988, Aug. 2012.

[22] Lingyao Kong and Jiadong Zang. Dynamics of an insulating skyrmion under a temperaturegradient. Phys. Rev. Lett., 111:067203, Aug 2013.

[23] N S Kiselev, A N Bogdanov, R Schfer, and U K Roßler. Chiral skyrmions in thin magneticfilms: new objects for magnetic storage technologies? Journal of Physics D: Applied Physics,44(39):392001, 2011.

[24] Albert Fert, Vincent Cros, and Joao Sampaio. Skyrmions on the track. Nature Nanotechnology,8(3):152–156, Mar. 2013.

[25] Niklas Romming, Christian Hanneken, Matthias Menzel, Jessica E. Bickel, Boris Wolter, Kirstenvon Bergmann, Andre Kubetzka, and Roland Wiesendanger. Writing and deleting singlemagnetic skyrmions. Science, 341(6146):636–639, 2013.

[26] R. Tomasello, E. Martinez, R. Zivieri, L. Torres, M. Carpentieri, and G. Finocchio. A strategyfor the design of skyrmion racetrack memories. Sci. Rep., 4:6784, Oct. 2014.

[27] Wataru Koshibae, Yoshio Kaneko, Junichi Iwasaki, Masashi Kawasaki, Yoshinori Tokura, andNaoto Nagaosa. Memory functions of magnetic skyrmions. Japanese Journal of AppliedPhysics, 54(5):053001, 2015.

[28] Xichao Zhang, Motohiko Ezawa, and Yan Zhou. Magnetic skyrmion logic gates: conversion,duplication and merging of skyrmions. Sci. Rep., 5:9400, Mar. 2015.


[29] Christian Hanneken, Andr Kubetzka, Kirsten von Bergmann, and Roland Wiesendanger.Pinning and movement of individual nanoscale magnetic skyrmions via defects. New Journalof Physics, 18(5):055009, 2016.

[30] I. Dzyaloshinskii. A thermodynamic theory of weak ferromagnetism of antiferromagnetics.Journal of Physics and Chemistry of Solids, 4(4):241 – 255, 1958.

[31] Toru Moriya. Anisotropic superexchange interaction and weak ferromagnetism. Phys. Rev.,120:91–98, Oct 1960.

[32] A. Fert and Peter M. Levy. Role of anisotropic exchange interactions in determining theproperties of spin-glasses. Phys. Rev. Lett., 44:1538–1541, Jun 1980.

[33] P Bak and M H Jensen. Theory of helical magnetic structures and phase transitions in mnsiand fege. Journal of Physics C: Solid State Physics, 13(31):L881, 1980.

[34] S. V. Maleyev. Cubic magnets with dzyaloshinskii-moriya interaction at low temperature. Phys.Rev. B, 73:174402, May 2006.

[35] R. Rajaraman. Solitons and Instantons, Volume 15: An Introduction to Solitons and Instantonsin Quantum Field Theory (North-Holland Personal Library). North Holland, 1 edition, 41987.

[36] Naoto Nagaosa and Yoshinori Tokura. Emergent electromagnetism in solids. Physica Scripta,2012(T146):014020, Jan. 2012.

[37] T. Schulz, R. Ritz, A. Bauer, M. Halder, M. Wagner, C. Franz, C. Pfleiderer, K. Everschor,M. Garst, and A. Rosch. Emergent electrodynamics of skyrmions in a chiral magnet. NaturePhysics, 8(4):301–304, Feb. 2012.

[38] N. Nagaosa, X. Z. Yu, and Y. Tokura. Gauge fields in real and momentum spaces in magnets:monopoles and skyrmions. Philosophical Transactions of the Royal Society of London A:Mathematical, Physical and Engineering Sciences, 370(1981):5806–5819, Nov. 2012.

[39] Xiao-Xiao Zhang, Andrey S. Mishchenko, Giulio De Filippis, and Naoto Nagaosa. Electrictransport in three-dimensional skyrmion/monopole crystal. Phys. Rev. B, 94:174428, Nov2016.

[40] P. Milde, D. Kohler, J. Seidel, L. M. Eng, A. Bauer, A. Chacon, J. Kindervater, S. Muhlbauer,C. Pfleiderer, S. Buhrandt, C. Schutte, and A. Rosch. Unwinding of a skyrmion lattice bymagnetic monopoles. Science, 340(6136):1076–1080, 2013.

[41] G E Volovik. Linear momentum in ferromagnets. Journal of Physics C: Solid State Physics,20(7):L83, 1987.

[42] P. R. Kotiuga. The algebraic topology of bloch points. IEEE Transactions on Magnetics,25(5):3476–3478, Sep 1989.

[43] N. Kanazawa, Y. Nii, X.-X. Zhang, A. S. Mishchenko, G. De Filippis, F. Kagawa, Y. Iwasa,N. Nagaosa, and Y. Tokura. Critical phenomena of emergent magnetic monopoles in a chiralmagnet. Nature Communications, 7:11622, may 2016.

[44] Rina Takashima and Satoshi Fujimoto. Electrodynamics in skyrmions merging. Journal of thePhysical Society of Japan, 83(5):054717, 2014.

[45] Filipp N. Rybakov, Aleksandr B. Borisov, Stefan Blugel, and Nikolai S. Kiselev. New type ofstable particlelike states in chiral magnets. Phys. Rev. Lett., 115:117201, Sep 2015.

[46] Christoph Schutte and Achim Rosch. Dynamics and energetics of emergent magnetic monopolesin chiral magnets. Phys. Rev. B, 90:174432, Nov. 2014.

[47] Shi-Zeng Lin and Avadh Saxena. Dynamics of dirac strings and monopolelike excitations inchiral magnets under a current drive. Phys. Rev. B, 93:060401, Feb 2016.

[48] M. N. Wilson, A. B. Butenko, A. N. Bogdanov, and T. L. Monchesky. Chiral skyrmions in cubichelimagnet films: The role of uniaxial anisotropy. Phys. Rev. B, 89:094411, Mar 2014.

[49] A. B. Butenko, A. A. Leonov, U. K. Roßler, and A. N. Bogdanov. Stabilization of skyrmiontextures by uniaxial distortions in noncentrosymmetric cubic helimagnets. Phys. Rev. B,82:052403, Aug 2010.

[50] H Wilhelm, M Baenitz, M Schmidt, C Naylor, R Lortz, U K Roßler, A A Leonov, and A NBogdanov. Confinement of chiral magnetic modulations in the precursor region of fege.Journal of Physics: Condensed Matter, 24(29):294204, 2012.

[51] B. Binz and A. Vishwanath. Theory of helical spin crystals: Phases, textures, and properties.Phys. Rev. B, 74:214408, Dec. 2006.

[52] Jin-Hong Park and Jung Hoon Han. Zero-temperature phases for chiral magnets in threedimensions. Phys. Rev. B, 83:184406, May 2011.

[53] N. Kanazawa, Y. Onose, T. Arima, D. Okuyama, K. Ohoyama, S. Wakimoto, K. Kakurai,S. Ishiwata, and Y. Tokura. Large topological hall effect in a short-period helimagnet MnGe.Phys. Rev. Lett., 106:156603, Apr. 2011.

[54] N. Kanazawa, J.-H. Kim, D. S. Inosov, J. S. White, N. Egetenmeyer, J. L. Gavilano, S. Ishiwata,


Y. Onose, T. Arima, B. Keimer, and Y. Tokura. Possible skyrmion-lattice ground state inthe B20 chiral-lattice magnet MnGe as seen via small-angle neutron scattering. Phys. Rev.B, 86:134425, Oct. 2012.

[55] Y. Shiomi, N. Kanazawa, K. Shibata, Y. Onose, and Y. Tokura. Topological nernst effect in athree-dimensional skyrmion-lattice phase. Phys. Rev. B, 88:064409, Aug 2013.

[56] Seong-Gyu Yang, Ye-Hua Liu, and Jung Hoon Han. Formation of a topological monopole latticeand its dynamics in three-dimensional chiral magnets. Phys. Rev. B, 94:054420, Aug 2016.

[57] Toshiaki Tanigaki, Kiyou Shibata, Naoya Kanazawa, Xiuzhen Yu, Yoshinori Onose, Hyun SoonPark, Daisuke Shindo, and Yoshinori Tokura. Real-space observation of short-period cubiclattice of skyrmions in MnGe. Nano Letters, 15(8):5438–5442, 2015. PMID: 26237493.

[58] Y. Nii, A. Kikkawa, Y. Taguchi, Y. Tokura, and Y. Iwasa. Elastic stiffness of a skyrmion crystal.Phys. Rev. Lett., 113:267203, Dec. 2014.

[59] A. E. Petrova and S. M. Stishov. Field evolution of the magnetic phase transition in the helicalmagnet mnsi inferred from ultrasound studies. Phys. Rev. B, 91:214402, Jun 2015.

[60] Gerald D. Mahan. Many-Particle Physics (Physics of Solids and Liquids). Springer, 3rd ed.2000 edition, 10 2000.

[61] Olga Petrova and Oleg Tchernyshyov. Spin waves in a skyrmion crystal. Phys. Rev. B,84:214433, Dec 2011.

[62] Takashi Koretsune, Naoto Nagaosa, and Ryotaro Arita. Control of dzyaloshinskii-moriyainteraction in Mn1−xFexGe: a first-principles study. Sci. Rep., 5:13302, Aug. 2015.

[63] J. Gayles, F. Freimuth, T. Schena, G. Lani, P. Mavropoulos, R. A. Duine, S. Blugel, J. Sinova,and Y. Mokrousov. Dzyaloshinskii-moriya interaction and hall effects in the skyrmion phaseof Mn1−xFexGe. Phys. Rev. Lett., 115:036602, Jul. 2015.

[64] K. Shibata, X. Z. Yu, T. Hara, D. Morikawa, N. Kanazawa, K. Kimoto, S. Ishiwata, Y. Matsui,and Y. Tokura. Towards control of the size and helicity of skyrmions in helimagnetic alloysby spin–orbit coupling. Nature Nanotechnology, 8(10):723–728, Sep. 2013.

[65] K. Shibata, J. Iwasaki, N. Kanazawa, S. Aizawa, T. Tanigaki, M. Shirai, T. Nakajima,M. Kubota, M. Kawasaki, H. S. Park, D. Shindo, N. Nagaosa, and Y. Tokura. Largeanisotropic deformation of skyrmions in strained crystal. Nature Nanotechnology, 10(7):589–592, Jun. 2015.

[66] Y. Hu and B. Wang. Unified Theory of Magnetoelastic Effects in B20 compounds. ArXive-prints, 1604.02766, April 2016.

arxiv:1608.06362v2 [cond-mat.mes-hall] 10 mar 2017 the monopole defects … · 2018-09-23 · with...

Documents